Multiscale Modeling of Polymer Composites: From Atomistic Simulation to Structural Analysis

Takaya Kobayashi, Kensuke Ogawa Mechanical Design & Analysis Corporation

Satoru Yamamoto, Riichi Kuwahara Dassault Systemes BIOVIA K.K.

Ryosuke Matsuzaki Tokyo University of Science

Abstract: A multiscale modeling procedure was presented to design thermosetting resins and products of polymer composites. Heat curing and cross-linking reaction steps were considered with activation energy and heat generation via molecular dynamics simulation of BIOVIA . The mechanical properties and adhesive strength for fillers were also estimated, and were assigned to . Using the Abaqus fracture analysis capability, the crack propagation behavior, including matrix-failure and interface-failure, was investigated.

Keywords: Polymer Composites, Multiscale Modeling, Atomistic Simulation and Structural Analysis.

1. Introduction

Polymer composites formed by thermosetting resins and carbon fibers are promising materials in transportation systems, such as airplanes and automobiles, owing to their light weight and high stiffness. In thermosetting resins, combinations of epoxy and amine compounds are widely used as the base resin and curing agent, respectively. The curing process of the thermosetting resin was investigated, as shown in Figure 1. The mechanical behavior of the curing process can be characterized by thermo-rheological behaviors (typically, the expression of elasticity owing to gelling and the volumetric change owing to curing shrinkage). From an experimental point of view, the expression of elasticity is evaluated by viscoelastic measurements using a shear rheometer with parallel plates, whereas the curing shrinkage is measured typically by volumetric dilatometry. These measurements must be followed by an estimation of the curing (hardening) degree, which indicates the degree of progression of the curing reaction of the polymeric materials. For this purpose, differential scanning calorimeter (DSC) measurements provide the degree of curing, and the Kamal differential equation is a typical application to describe the progress of the curing reaction of the 1

thermosetting resin. Abaqus allows introduction of the Kamal model as a user-defined state variable, which makes it possible to provide the curing degree dependencies on the elastic modulus and the curing shrinkage. There are many varieties of epoxy and amine compounds, and their choice is a key strategy for material design because their combinations as well as curing conditions strongly affect the mechanical and thermal properties. The conventional trial-and-error procedures are expensive and time-consuming, and therefore, computer modeling is desired in order to reduce cost and accelerate the development. With this background, we propose a multiscale modeling procedure (Figure 2) to design an epoxy resin and a polymer composite. Our approach consists of two parts: an atomistic simulation for curing reaction and estimation of the mechanical and thermal properties, and an FEM simulation for the molding process and structure. In the first part, a heat curing reaction is simulated by the molecular dynamics (MD) method in order to explore how combinations of the base resin and curing agent affect the cross-linked structure and material properties. Mechanical and thermal properties of the epoxy resin and the adhesive strength between the resin and fillers are also predicted via MD simulation. In the FEM part, first, the distribution of the conversion of the curing reaction throughout the product is simulated. By assigning the obtained material properties to each part of the product in reference to the conversion of curing reaction, the stress and crack propagation are investigated for a part of the product and the whole product of the polymer composite. These approaches are expected to provide the prospect of solving the material irregularity in the actual polymer composite as shown in Figure 3 (Yoshimura, 2016).

Figure 1. Experimental approaches to curing process of thermosetting resin.

2

Figure 2. Multiscale modeling of polymer composites.

Figure 3. Irregularities in polymer composite.

2. Atomistic simulation

The reaction of epoxy compounds with primary amines takes place through the following two steps:

(1)

3

(2)

When the epoxy group of the base resin approaches the amino group of the curing agent, a cross- link is constructed and a secondary amine is produced. The secondary amine reacts again with the epoxy compounds and produces a ternary amine. In this paper, we choose two epoxy compounds, DGEBA (Diglycidyl ether of bisphenol A)

and TGDDM (Tetraglycidyl diaminodiphenylmethane).

The number of epoxy groups is different in DGEBA and TGDDM, i.e., 2 and 4, respectively. For the amine compound, we choose 44DDS (4’4-Diaminodiphenylsulphone).

The following calculations are performed by using the software package Materials Studio 2017 R2 (Dassault Systèmes BIOVIA). The Amorphous Cell module is used to construct atomistic models for two systems in stoichiometry; (a) 40 DGEBA and 20 44DDS, and (b) 40 TGDDM and 40 44DDS. The total number of atoms in the system is (a) 2,540 and (b) 3,600. The curing simulation is performed with the Forcite module and the COMPASSII force fields according to the molecular dynamics technique proposed by Okabe et al. (Okabe, 2016). After relaxation of the system at NPT ensemble, MD simulation is carried out to reproduce the cross- linked structure of the epoxy resin. When the epoxy group approaches to the amino group within the reaction range, the reaction probability k, which consists of the acceleration factor A and the activation energy G in Equation 3, is compared with a random number P (0-1) 4

∆퐺 k= 퐴 푒푥푝 (− ) (3) 푅푇 where R is the gas constant and T is the local temperature. In the acceptable condition, the curing reaction occurs, and the cross-linked structure is generated. After the chemical reaction, the heat of formation Hf is considered by increasing the kinetic energy of the reacted part. The heat of formation increases the local temperature and accelerates the following reactions. The activation energy and heat of formation are summarized in Table 1. Both the activation energy of TGDDM and 44DDS are slightly higher than those of DGEBA and 44DDS. Figure 4 shows the evolution of the cross-linked structures at 473 K. An isolated structure is indicated by a different color. It is clearly understood that the cross-linked structure increased with the conversion of the reaction. In case of DGEBA/44DDS, the percolated network emerged at 95% conversion. However, full percolation occurred at 65% conversion in the case of TGDDM/44DDS owing to the difference in the number of epoxy groups. From these simulations, we can understand how the cross-linking reaction proceeds. Figure 5 shows a change in the different amino groups as a function of conversion. The ratio of the secondary amino group increases to reach its maximum value 0.5 at about 50% conversion in the case of DGEBA/44DDS. This means that the rates of production and consumption of the secondary amino groups are equivalent, and this tendency agrees with the experimental observation via near infrared spectroscopy (Min, 1993). The ratio of the secondary amino groups of TGDDM/44DDS, however, shows asymmetric change for conversion reaching its maximum value 0.35 at about 40% conversion. It is supposed that the second reaction in Equation 2 is slower than the first one in Equation 1. This difference is also caused by the number of epoxy groups and the chemical structure and not by the energetic barrier because the 1st and 2nd activation energy have the same magnitude. After cooling the cross-linked structure to 298 K, for each direction (x, y, z), a number of strains (0–0.1) are applied by MD simulation, resulting in a strained structure. Young’s modulus is estimated by averaging the stress–strain relationship for all directions, as shown in Table 2. The value of Young’s modulus tends to increase as the conversion increases, and this tendency is more noticeable and Young’s modulus is much higher at a high conversion for TGDDM/44DDS. Thermal properties, i.e., the coefficient of thermal expansion and the glass transition temperature, can be obtained by monitoring the relationship between the volume and temperature of the system. Next, the interfacial adhesion strength between a filler and epoxy resin is calculated. Here, we consider a copper substrate with (111) free surface as a filler, because the surface of carbon fiber is not fully clarified and a previous study has investigated a copper surface by MD (Yang, 2010). If an atomistic structure of the carbon fiber surface is clearly understood, we can create its model and perform similar simulations. A mixture of epoxy and amine compounds is placed over the copper surface, and then, the curing reaction is simulated to obtain the initial structures for peel-off simulation. Periodic boundaries are considered in the system, and therefore, it is assumed to be a sandwich structure of epoxy resin between the upper and lower copper layers. When the copper layer is fixed, a small displacement is applied in perpendicular to the copper surface and the stress is monitored via MD simulation. Two distinctive snapshots are shown in Figure 6 for 65% and 95% conversion for TGDDM/44DDS. Note that strain 1 refers to 5.7 nm displacement. In case of 65% conversion, a void is initiated within the epoxy resin at strain 0.2, and the voids grew in size and several chain segments of the cross-linked network become stretched (Strain 0.6 and 1.0). 5

Even at strain 1.0, the epoxy resin adheres to the copper surfaces. In contrast, the epoxy resin is peeled off at the interface in case of 95% conversion. This may be caused by the high rigidity of epoxy resin. The corresponding stress–strain relationships are shown in Figure 7. At high conversion of 80% and 95%, the yield stress is clearly recognized owing to peel-off at the interface. Under 65% conversion, however, after yield stress at about strain 0.1, plastic deformation is observed, and this means cohesive failure in epoxy resin.

Figure 4. Evolution of cross-linked structure.

Figure 5. Change of different amino group ratios through the reaction. 6

Figure 6. Snapshot of peel-off simulation (TGDDM/44DDS).

Figure 7. Stress–strain relationship of peel-off simulation (TGDDM/44DDS).

7

Table 1. Activation energy and heat of formation.

Combination G (kcal/mol) Hf (kcal/mol) 1st 2nd 1st 2nd DGEBA/44DDS 42.01 40.79 28.31 22.05 TGDDM/44DDS 43.68 43.81 26.07 28.87

Table 2. Young’s modulus of curing products.

Young’s modulus (GPa) Conversion (%) DGEBA/44DDS TGDDM/44DDS 35 0.97 1.05 50 1.38 1.38 65 1.63 1.78 80 1.99 2.03 95 1.77 4.36

3. Structural Analysis

Figure 8 shows the multiscale modeling of polymer composites in this study, which is a combination of BIOVIA Materials Studio and SIMULIA Abaqus. As shown in Figure 6 and Figure 7, Materials Studio successfully provided the microscale stress–strain curves suggesting fracture mechanisms on the molecular level. Since fracture strength in the atomistic simulation can be recognized as a kind of ideal strength of materials, the difference from actual strength should be considered carefully. However, the analytical results of Materials Studio, which surely distinguish between a matrix-failure and an interface-failure, will be significantly useful in the selection of material in the polymer composite design. In the FEM part, SIMULIA Abaqus was used for its powerful nonlinear capability for fracture modeling (Fukui, 2016). The atomistic simulation results were applied so as to conform to the provisions of the Abaqus fracture analysis capabilities. Figure 9 shows the finite element model, which is a periodic unit cell model containing randomly positioned 12 fibers in the matrix. A 2D plane strain condition was assumed, whereas the finite element mesh represented by the solid element C3D8 was pseudo three-dimensional in anticipation of practical needs for three- dimensional fracture analysis. The matrix and the fiber were assumed to be elastic, whose Young’s moduli were 2,480 MPa and 298,000 MPa, respectively. The Poison’s ratio was assumed to be 0.3 in common.

In this study, the characteristics of transverse failure in a unidirectional composite are simulated. A transverse crack, the so-called first-ply failure, causes considerable damage and can trigger fatal delamination. A numerical simulation of the progressive failure growth, consisting of initiation of transverse matrix cracking, crack propagation, and debonding of fiber–matrix interface, was demonstrated in this study. As shown in Figure 9, the eXtended finite element method (XFEM) 8

was used to simulate the matrix-failure. As for the fiber–matrix interface, 8-node cohesive elements (COH3D8) were prescribed to simulate the delamination behavior with the cohesive zone model (CZM). The cohesive elements were modeled with 0.5 μm thickness to avoid interpenetration. Figure 10 shows the multiscale modeling of the interface-failure in this study. During the interface crack growth process, two new surfaces are created. Before the crack is formed, these two surfaces can be assumed to be held together by traction to compose the interface. The CZM provides a traction-separation law for the gradual fracture formation took place in the local area (so-called cohesive zone, or fracture process zone) ahead of the crack-tip. As the surfaces separate, traction 0 first increases until a maximum tn is reached, and then reduces to zero, which results in complete separation. The area under this curve is equal to the energy needed for separation Gc. Namely, The CZM assumes that the failure of the cohesive elements is characterized by progressive degradation of the material stiffness, which is driven by a damage process. As the maximum stress is limited to the cohesive strength of the material, this technique provides a stable solution for interface-failure problems, which tend to be singular by physical discontinuity due to debonding. As shown in Figure 10, if the “peel-off” cases in the microscale stress–strain curves obtained by Materials 0 Studio are equated with an interface-failure, the maximum traction tn and the critical energy release rate Gc can be identified from the stress–strain curves. Figure 11 shows the multiscale modeling of the matrix-failure using XFEM. The XFEM algorithm allows the presence of discontinuities such as cracks in an element by enriching the degrees of freedom with special displacement functions. As the crack tip changes its position, XFEM creates the necessary enrichment functions for the nodal points around the crack tip. Therefore, the technique is a very attractive way to simulate initiation and propagation of a discrete crack along an arbitrary, solution-dependent path without the requirement of remeshing in the materials. Similar to the above-mentioned idea, if the “void creation” cases in the microscale stress–strain curves obtained by Materials Studio are equated with a matrix-failure, the crack initiation and its growth can be identified from the stress–strain curves. The maximum principal stress criterion and the linear elastic fracture mechanics criterion were applied to the initiation and the propagation criteria, respectively. As mentioned above, the fracture strength obtained in the atomistic simulation is close to the ideal strength of materials. Naturally, the practical strength for both interface-failure and matrix-failure depends on a number of factors. We have performed a comprehensive numerical study for the failure strength in consideration of the quantitative results from the atomistic simulation of Figure 10 and Figure 11. Figure 12 shows a typical result of our analytical study for the relationship between nominal stress n and nominal strain n. The term “nominal” refers to the cross-sectional average of the stress and strain. Figure 12 also shows deformed shapes obtained from the analysis. It is helpful to review the process of failure by following the order of events. First, an initial crack occurs at the narrowest part of the matrix between two fibers (n = 0.0001), where the maximum of the maximum principal stress is observed. In this portion, the deformation of the matrix is localized because this matrix area is held between two rigid fibers and given a forced displacement. The initial crack starts growing in accordance with the critical energy release rate Gc but the nominal stress increases as long as the failure is limited in the small part of the model (n < 0.0012). Next, the crack propagates rapidly, and the nominal stress sharply decreases (n = 0.0015). 9

However, when the crack tip hits the outer surface of fiber, the propagation stops. This behavior is referred as a crack arrest, in which an unstable crack extension may start in a region of high stress, but that the surrounding material will have sufficient resistance to the crack extension to arrest the running crack. In fact, after the crack arrest, the nominal stress increases (n < 0.0031), until an interface-failure occurs. As the fibers are positioned randomly in the matrix, other interface- failures take place successively (n = 0.0040). Along with them, the load drops rapidly, yielding the loss of the load-bearing capacity (n = 0.0057).

Figure 8. Multiscale modeling of polymer composite.

Figure 9. Periodic unit cell model containing 12 fibers in the matrix. 10

Figure 10. Multiscale modeling of interface-failure.

Figure 11. Multiscale modeling of matrix-failure.

11

Figure 12. Failure behaviors transiting from matrix-failure to interface-failure.

4. Conclusions

A multiscale modeling of a polymer composite, which is a combination of BIOVIA Materials Studio and SIMULIA Abaqus, was demonstrated. The microscale stress–strain curves suggesting fracture mechanisms of the molecular level were assigned to the finite element analysis, and the failure behaviors transiting from matrix-failure to interface-failure were analyzed. The study showed the potential of the multiscale analysis to improve the quantitative aspect in design and material selection of the polymer composite, especially with respect to failure characteristics of the material, which cannot be easily grasped by material testing alone.

5. References

1. Okabe, T., Oya, Y., Tanabe, K., Kikugawa, G., and Yoshioka, K., “Molecular Dynamics Simulation of Crosslinked Epoxy Resins: Curing and Mechanical Properties” European Polymer Journal 80, 78, 2016.

12

2. Min, B.G., Stachurski, Z.H., Hodgkin, J.H., and Heath, G.R., “Quantitative Analysis of the Cure Reaction of DGEBA/DDS Epoxy Resins without and with Thermoplastic Polysulfone Modifier using Near Infra-red Spectroscopy” Polymer 34, 3620, 1993. 3. Fukui, H., Yoshimura, A., and Matsuzaki, R., “Structural optimization for CFRP cryogenic tank based on energy release rate” Composite Structures, 152-15, 883, 2016. 4. Yang, A., Gao, F., and Qu, J., “A study of highly crosslinked epoxy molding compound and its interface with copper substrate by molecular dynamics simulations” 2010 Electronic Components and Conference, 128, 2010. 5. Yoshimura, A., Waas, A.M., Fukui, H., Nakayama, M., and Matsuzaki, R., “Multiscale analysis of stitched CFRP composites including the effect of geometrical imperfection” 31st ASC Technical Conference and ASTM D30 Meeting, 1004, 2016.

13